Question: Match the polynomial on the left with the appropriately factored expression on the right. $4x^2 + 18x$ $4x^2 + 2x + 18x + 9$ $2x(2x + 1)$ $2x(2x + 9)$ $(2x + 1)(2x + 9)$ $(2x + 3)(2x + 3)$

1. **State the problem:** We need to match each polynomial on the left with its correct factored form on the right. 2. **Factor $4x^2 + 18x$:** - Find the greatest common factor (GCF) of $4x^2$ and $18x$, which is $2x$. - Factor out $2x$: $$4x^2 + 18x = 2x(2x + 9)$$ 3. **Factor $4x^2 + 2x + 18x + 9$:** - Combine like terms inside the polynomial: $2x + 18x = 20x$, but since the expression is given as four terms, we use grouping. - Group terms: $$ (4x^2 + 2x) + (18x + 9) $$ - Factor each group: $$ 2x(2x + 1) + 9(2x + 1) $$ - Factor out the common binomial factor $(2x + 1)$: $$ (2x + 1)(2x + 9) $$ 4. **Match each polynomial with its factored form:** - $4x^2 + 18x$ matches $2x(2x + 9)$. - $4x^2 + 2x + 18x + 9$ matches $(2x + 1)(2x + 9)$. 5. **Check other options:** - $2x(2x + 1)$ does not match either polynomial. - $(2x + 3)(2x + 3)$ expands to $4x^2 + 12x + 9$, which does not match any given polynomial. **Final matches:** - $4x^2 + 18x$ $ o$ $2x(2x + 9)$ - $4x^2 + 2x + 18x + 9$ $ o$ $(2x + 1)(2x + 9)$